Research Article Vol. 34, No. 5 / May 2017 / Journal of the Optical Society of America B 937
Stimulated Brillouin scattering in integrated ring resonators
SAYYED REZA MIRNAZIRY,1,2,* CHRISTIAN WOLFF,1,2 M. J. STEEL,1,3 BENJAMIN J. EGGLETON,1,4 AND CHRISTOPHER G. POULTON1,2
1Centre for Ultrahigh bandwidth Devices for Optical Systems (CUDOS), Australia
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2School of Mathematical and Physical Sciences, University of Technology Sydney, Ultimo, NSW 2007, Australia
3MQ Photonics Research Centre, Department of Physics and Astronomy, Macquarie University, Sydney, NSW 2109, Australia 4Institute of Photonics and Optical Science (IPOS), School of Physics, University of Sydney, Sydney, NSW 2006, Australia *Corresponding author:
Received 31 January 2017; accepted 6 March 2017; posted 10 March 2017 (Doc. ID 285828); published 13 April 2017
We investigate stimulated Brillouin scattering (SBS) in ring resonators that exhibit linear and nonlinear losses. We present both analytic and numerical tools that can be used to compute the amplification and lasing threshold for the SBS-induced Stokes line. We show that, for both linear and nonlinear losses, there is a maximum achiev- able SBS amplification, and we show how this depends on the parameters of the ring and on the material param- eters. We also study the relation between the critical coupling and input pump power to achieve amplification in various situations. We present simplified models that can be used to accurately predict the nonlinear behavior of the ring and consequently estimate the pump power required to achieve the optimum gain in a range of technologically important situations. © 2017 Optical Society of America
OCIS codes: (190.0190) Nonlinear optics; (190.2640) Stimulated scattering, modulation, etc.; (230.1040) Acousto-optical devices. https://doi.org/10.1364/JOSAB.34.000937
1. INTRODUCTION
Stimulated Brillouin scattering (SBS) is a coherent opto- mechanical interaction by which light in a narrow-linewidth pump can be transferred to a closely spaced Stokes signal by a high-frequency sound wave [1]. SBS has attracted much attention recently in the context of integrated photonic wave- guides [2]; the motivating applications include optical signal processing [3], compact broadband microwave photonic devi- ces [4], and compact narrow-linewidth lasers [5–8]. Various platforms have been explored or proposed to harness SBS; these include photonic crystal fibers [9,10], nanowires [11], slot waveguides [12], microdisks [6], and ring resonators [8,13]. Other platforms, such as optical fibers, have also been investi- gated in the area of the SBS lasers [14].
In most integrated SBS-based applications it is desirable to achieve a strong output Stokes power within a very small physi- cal footprint. In principle, this can be achieved using long lengths of folded integrated waveguides, operated at high power to overcome losses from bending and roughness. An attractive alternative is to use a high Q-factor resonator, with either the pump, the Stokes, or both tuned to coincide with the resonan- ces. High-Q ring resonators have long been used in photonics to enhance nonlinear effects [15], and these structures are also beginning to be used in SBS experiments [6,13,16], as they have the potential to drastically reduce the required input
pump power [17]. Other resonators, such as phase shifted gratings, have also been proposed to demonstrate distributed feedback Brillouin lasers [18]. However, in many technologi- cally interesting material platforms it is likely that nonlinear losses, such as two-photon absorption (TPA) and free-carrier absorption (FCA), will play a major role in the SBS response. Indeed, in straight waveguides it has been shown that nonlinear losses result in a strict upper bound for the achievable Stokes amplification [12,19], and a similar situation is likely to occur in rings. In addition, a complete study of the physics of SBS in ring resonators, even to the minimal extent of including linear optical loss, does not yet exist in the literature. A proper quan- titative understanding of SBS in ring resonators, including the important effects of all loss terms, is necessary for these struc- tures to be viable in SBS-based integrated devices.
In this paper, we study the SBS response of ring resonators acting as Stokes amplifiers in the steady state. We examine the effects of linear and nonlinear optical losses, and investigate the physics of the interaction between linear and nonlinear effects. To this end we present several models that are appropriate for investigating SBS in rings, ranging from the fully numerical to the analytic, and compute the net Stokes amplification that can be achieved in technologically interesting cases. We identify different response regions of the ring and present expressions for threshold input pump power required for SBS lasing.
0740-3224/17/050937-13 Journal © 2017 Optical Society of America
938 Vol. 34, No. 5 / May 2017 / Journal of the Optical Society of America B
Research Article
We find that, similar to straight waveguides, there is a maxi- mum Stokes power that can be obtained from a given ring, and we present tools for calculating and estimating the optimum pump power required to obtain maximum Stokes gain. Where possible we present results in terms of generalized parameters, common to all rings, that can be used as a basis for resonator design and for finding the optimum operating point in experiments. Finally, we investigate the impact of variation of nonlinear dispersion and ring length on the free spectral range (FSR) of the ring, and hence on the SBS response; in Appendix A we introduce criteria to demonstrate the acceptable tolerance of material and waveguide parameters to prevent reduction in the output Stokes power.
2. GOVERNING EQUATIONS
We consider the geometry shown in Fig. 1, which consists of a
single guiding loop of length L connected to a bus waveguide
via a coupling region of length Lc. Since Lc is considerably
shorter than L, we neglect the impact of loss in the coupling
region. Pump and Stokes waves of complex amplitude Ain and p
Ain are input from the left- and right-hand sides of the ring, s
respectively. Although the notation in this paper is appropriate for both backward and forward SBS, the definitions proposed for Stokes transmission and nonlinear loss coefficients are only applicable to backward SBS. We assume that the coupling be- tween the ring and the loop can be described by coupling co- efficients κ1;2, which denote the cross-coupling coefficients from the waveguide to the ring and vice versa, and τ1;2, which characterize the transmission through the coupler. Finding the coupling coefficients has been the subject of several numerical and experimental studies [20]; here we assume that these co- efficients are constant within the operating frequency range. Moreover, we assume that the waveguide geometry is suffi- ciently symmetric and lossless in the coupling region that τ1 τ2 and that κ1 κ2. Then, the relationship between the coupling coefficients is given by
τ21−κ21e2iφτ; (1)
where φτ is the phase of the transmission coefficient τ1. The relationship between the φτ and the cross coupling phase term φκ is then
φτ − φκ π : (2) 2
Within the ring we define the optical fields of the pump and Stokes waves as functions of the coordinate z, which labels the clockwise distance around the ring as shown in Fig. 1, as
Epz BpzE ̃p;r c:c:; (3) Esz BszE ̃s;r c:c:; (4)
where E ̃p;r and E ̃s;r are the fields associated with the pump and Stokes eigenmodes, respectively, of the ring, and the optical envelopes Bpz and Bsz are defined to include the optical phase:
Bpz 0 Aout !
κ2eiK p;rLc τ eiK L
τ2eiK p;wLc κ eiK L !
Bpz L Ain !
BpzB ̃p expiKp;rz; (5) ̃
Bsz Bs exp−iK s;rz; (6)
in which B ̃p and B ̃s are non-phase-dependent, slowly varying functions of the variable z within the ring. Here K p;r (K s;r) and K p;w (K s;w ) are propagation constants of pump (Stokes) in the ring and bus waveguide, and are labeled by either r (ring) or w (waveguide) as appropriate.
The optical envelopes in the bus waveguide and ring are related in the coupling region to the output envelopes by matrix equations [21] ! ! !
Aout τ eiKp;wLc κ eiKp;r p11p;
s 1s;wc 1s;rc Bsz L κ2eiK s;rLc τ2eiK s;wLc
(7) s ;(8)
where Aout and Aout are the output envelopes, defined at the
left-hand-side end of the coupler (for Stokes) and at the right- hand-side end of the pump. We note that the optical phase variation of the modes in the coupling region is separated from the coupling coefficients. Therefore, κi and τi (i 1; 2) can be complex valued.
From Eqs. (7) and (8) the total power transmission can be defined for both pump and Stokes:
Aout2 jτ j−D eiφτφp;rKp;rLc 2
p Ain 1−D
p;r ; (9) jτ jeiφτφp;rKp;rLc
where φp;r (φs;r) is the pump (Stokes) envelope phase change in a round trip from z 0 (z L) to z L (z 0), given by
and Dp;r and Gs;r are defined as the round-trip depletion and gain round trip:
Dp;r jB z 0j; (13) p
Aout2 jτ j − G eiφτφs;rK s;rLc 2
T s 1 s;r
s Ain 1−G jτ jeiφτφs;rKs;rLc
φp;r φs;r
Kp;rdz; (11) −Ks;rdz; (12)
Schematic of a ring resonator in the vicinity of a straight coupler. The length of the coupling region Lc is assumed to be considerably smaller than L.
jBpz Lj
Research Article
Vol. 34, No. 5 / May 2017 / Journal of the Optical Society of America B 939
G jBsz0j: s;r jBsz Lj
the pump and Stokes transmissions in Eqs. (9) and (10)
simplify to
jτ j − D 2
Tp 1 p;r; (19)
Ts 1 s;r: (20)
Efficient operation of a ring resonator for SBS applications occurs at a resonance in which both pump and Stokes ampli- tudes build up within the ring, and a given Stokes net gain can be achieved with minimum input pump power. The condition for achieving a resonance in the ring requires the phase change over a single round trip to satisfy
φpφτφp;rKp;rLc2Mπ; (15) φs φτ φs;r Ks;rLc 2Nπ; (16)
where M and N are integers. For maximum SBS gain, the length of a ring resonator should be designed such that both pump and Stokes lie on a resonance. In the absence of dispersion in Eqs. (11) and (12), the phase change in a single round trip is
φp;r Kp;rL; φs;r Ks;rL:
Then, by substituting to Eqs. (15) and (16) the length of ring for simultaneous pump-Stokes resonance can be obtained:
M − N 2π LRLLcK −K : (17)
The smallest length of a ring resonator occurs when the cor-
responding pump and Stokes frequencies are the consecutive resonances of the ring, i.e., when M − N 1. In this case we have
FSR f B; (18)
where FSR is the free spectral range and f B is the Brillouin frequency. This situation is shown in Fig. 2, where the pump and Stokes lie on the successive resonances of ring. For a ring resonator consisting of a conventional optical waveguide, such as a nanowire, rib, or slot waveguide, LR is about 1 cm for the Brillouin frequencies around 10 GHz. As long as Eq. (18), or the more general Eq. (17), is obeyed, the expressions for
1 − jτ2jDp;r
1 − jτ2jGs;r
From Eq. (19) the pump transmission tends to zero as
jτ1j Dp;r; (21)
which is the condition for critical coupling. We note that due to the effect of dispersion the resonance frequencies can be slightly shifted. We assume that the pump and Stokes frequencies are adjustable to keep them on resonance.
From Eqs. (19) and (20) it can be seen that the quantities Gs;r and Dp;r are important for finding overall transmission properties. These quantities, however, depend on the nonlinear interactions between the pump and Stokes fields within the ring itself. The governing equations for coupling between the pump and Stokes envelopes via SBS in the ring are [19]
∂Bα p− −iKp;rβp;2PAjBpzj2γpjBpzj4 Bpz
s − iK s;r − βs;2PAjBszj2 − γsjBszj4 Bsz
Here, α2 is the amplitude linear loss coefficient; βp;2PA (βs;2PA) is the coefficient for the third-order Kerr effect and TPA, and γp (γs) is the coefficient for the fifth-order nonlinear effects FCA and free-carrier dispersion (FCD). The transfer of optical power from the pump to the Stokes field is defined by the SBS gain parameter Γ0. Each of these coefficients appearing in Eqs. (22) and (23) has units of m−1 because the envelope functions are defined to be dimensionless; these coefficients are related to tabulated coefficients, β2PA, γ, and Γ, which depend on the optical power, by a factor of 2:
β2PA 2βp;2PA −2βs;2PA 1∕Wm;
γ 2γp −2γs 1∕mW2; Pnorm2 Pnorm2
Γ Pnorm 1∕mW;
where Pnorm 1 W has been introduced to make the quantities dimensionally correct.
3. SBS IN RING RESONATORS WITH LINEAR LOSS
We begin by studying SBS in ring resonators with only linear loss and negligible material dispersion in the steady state. This
−2βp;2PA 4γpjBpzj2 γpjBszj2 Γ0jBszj2Bpz;
− 2βs;2PA 4γsjBszj2 γsjBpzj2 Γ0jBpzj2Bsz:
Stokes pump
Illustration of the transmission spectrum of a ring resonator. Pump and Stokes are assumed to be on resonance (red arrows). The smallest length of a ring designed for SBS applications corresponds to the case that the pump and Stokes frequencies lie on consecutive resonances of the ring [see Eq. (17)].
Transmission
940 Vol. 34, No. 5 / May 2017 / Journal of the Optical Society of America B Research Article
is a good approximation for glasses, including chalcogenides, in which TPA is negligible and free carriers are nonexistent.
In the absence of higher order nonlinear losses, the governing equations for the pump and Stokes are given by [see Eqs. (22) and (23)]
is 10 GHz and that pump and Stokes lines fall on consecutive resonances. The Stokes amplification is significantly higher at the resonance frequency, when large optical power builds up within the ring. A small shift—Δλp ≈ 5 pm—from the reso- nance results in a significant reduction in the Stokes gain.
For SBS gain in a resonator, there is a minimum pump power for which the Stokes signal can be amplified. This power, which is required to exactly compensate the linear loss of the round trips through the resonator, can be obtained by analyti- cally solving Eqs. (24) and (25) [assuming that the pump depletion due to SBS is neglected in Eq. (24)] together with Eqs. (7) and (8) at the resonance frequency:
∂B α p − − iK p;r
A. SBS with Linear Loss
Bpz − Γ0jBszj2Bpz;
Bsz − Γ0jBpzj2Bsz:
∂B α s − iK s;r
By neglecting the effect of pump depletion from SBS it is pos- in;min αL α 1 − jτ1je−αL∕22
sible to obtain analytic expressions for the pump and 1 − e−αL Γ jκ1j2 : (29)
transmissions. This approximation is valid only in the regime of lower Stokes gain. By using this approximation to solve Eqs. (24) and (25) we obtain the following expressions for Dp;r and Gs;r [required to find the power transmission via Eqs. (9) and (10)]: !
Figure 3(b) shows the variation of the minimum required power given in natural units of α∕Γ for ring resonators at resonance. From the figure, it can be seen that the minimum pump power required to obtain a given Stokes amplification increases with the normalized loss. The solid-circle line shows the critical coupling. At the critical coupling, the pump power required to achieve a 0 dB Stokes transmission is minimized. We will see that increasing jκ1j is also advantageous in increas- ing the maximum achievable gain of the structure.
B. SBS with Linear Loss and Pump Depletion
As we increase the input power from its minimum value the magnitude of the Stokes field, both in the ring and in the out- put, will increase. In addition, the impact of the pump depletion term grows in Eq. (24); for large Stokes gains, this term is no longer negligible. We here make an approximation for the pump depletion loss term in Eq. (24) by considering an average value for the Stokes envelope in the ring:
p − iK p;r − Γ0jBs;avj2 Bpz: (30)
Dp;r exp −αL ; 2
Gs;rexp −2αjBpz0j21−e−αL;
where the pump field at z 0 is κ2eiKp;rLc
Bz0 Ain; (28) p 1 − jτ2je−α2Liφp p
in which we have neglected the propagation loss in the coupling region. Figure 3(a) shows an example of the Stokes transmission in a ring resonator with the linear loss coefficient α 34.54 m−1 (corresponding to a loss of 1.5 dB cm−1 ) and SBS gain Γ 400 W−1 m−1 for a range of input pump powers between 24 and 32 mW. The length of the ring is designed according to Eq. (17) assuming that the Brillouin frequency
Fig. 3. (a) Example of the Stokes amplification in a ring resonator for a range of input pump powers varying from 24 to 32 mW. The linear loss is
assumed to be 1.5 dB/cm; Γ 400 W−1 m−1; L and Lc are 1.745 cm and 7.476 μm, respectively; and jκ1j is 0.6. The impact of pump depletion is
ignored. (b) Contours of the Γ∕αPmin in natural units at the resonance frequency. It is assumed that the ring is designed according to the method p
demonstrated in Section 2.
Research Article Vol. 34, No. 5 / May 2017 / Journal of the Optical Society of America B 941
Note that we have not specified the quantity Bs;av at this stage. Now we assume that
κ1Bsz 02 Tsτ1pffiffiffiffiffiin ;
jτ jjB j ≫ jκ jjAin j; 2s 2s
2jκ j2jB j2
≈1s;av: (36)
which is satisfied when the Stokes envelope within the ring is significantly larger than the input Stokes. Then, from Eq. (8),
jBsz Lj2 ≈ jτ2j2jBsz 0j2:
Now we approximate jBs;avj2 as the linear interpolation
jBs;avj2 2 1 jτ2j2jBsz 0j2: (32)
Note that this approximation is valid for non-small values of jτ2j. We can solve Eq. (24) analytically to find the pump envelope Bp, as a function of input pump envelope, α, coupling coefficients, and jBs;avj. Then, by substituting the pump envelope in Eq. (25) and applying the boundary conditions in Eq. (8), we have
1 qffiffiffiffiffi
jB jeη −jτ j− jκ j R jAinj1eη0; (33)
1 jτ j2R jAinj2 2pp
s;av 221pp where η is given by
In solving Eq. (33) the initial guess for jBs;avj should be chosen carefully to avoid convergence to false roots.
Figure 4(a) shows the variation of the Stokes gain as a func- tion of input pump power for four case studies. The dashed lines show the Stokes transmission in the absence of pump depletion. In each case, T s first grows rapidly with increasing the pump power, then it levels off, which indicates a balance between the Stokes amplification and optical losses in the ring at large values of Stokes transmission. We see that for larger SBS gains, the pump power required to achieve the maximum Stokes amplification is reduced. For each case study there is a perfect match between the solid and dashed lines at smaller Stokes gains that shows the negligible effect of pump depletion.
As we further increase the input power, the dashed line transmission fails to accurately estimate the Stokes gain. As the Stokes transmission reaches its maximum, the output Stokes does not change by varying the initial Stokes. If, for in- stance, the initial Stokes increases, given the pump power to be constant, then from Eq. (35), Rp rises proportionally, which causes reduction in the Stokes transmission [see Eq. (36)]. Consequently, the output Stokes power remains fairly un- changed as the reduction in T s is compensated with increasing the initial Stokes. Therefore, the ring acts as an oscillator rather than as an amplifier. We here define the lasing region as the power interval at which the final values of the Stokes becomes less influenced by changing the pump power in the small signal approximation. The lasing threshold is then given by
in;th αL − 2 ln jτ2j α 1 − jτ1je−αL∕22
P 1−e−αL Γ
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